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3 Exponential and Logarithmic functions 3.1 Exponent Here are a few examples to remind the reader of the definitions and laws for expressions involving exponents: 23 = 2 · 2 · 2 = 8, 20 = 1, 1 2−1 = , 2 1 1 −3 2 = 3 = , 2 8 √ 91/2 = 9 = 3, √ 93/2 = ( 9)3 = 27 1 1 1 8−2/3 = 2/3 = √ = , 3 2 4 8 ( 8) a2 a3 = (aa)(aaa) = a5 = a2+3 , (a2 )3 = (aa)(aa)(aa) = a6 = a2·3 . The last two lines illustrate the following identities, which hold for any real numbers a, x and y with a > 0: Law of exponents. (i) ax ay = ax+y , (ii) (ax )y = axy . The seemingly strange definitions 20 = 1, 2−1 = 1/2, and so on are forced on us if we wish for the laws of exponents to be valid for all real numbers x and y as stated: 20 · 2 = 20 · 21 = 20+1 = 21 = 2 forces 2−1 · 2 = 2−1 · 21 = 2(−1)+1 = 20 = 1 3.2 20 = 1, forces 2−1 = 1 . 2 Exponential function The graph of the function f (x) = 2x can be sketched by plotting a few points and connecting: 1 3 EXPONENTIAL AND LOGARITHMIC FUNCTIONS x −2 −1 0 1 2 2 f (x) = 2x 1/4 1/2 1 2 4 From the discussion above, we know what is meant by f (x) when √ x is a rational number (i.e., fraction), since we have f (m/n) = 2m/n = ( n 2)m . But since we have not defined 2x when x is irrational, the√ graph above actually has a hole in it above every irrational number, such as 3. We remedy this situation by defining 2x for irrational x in such a way that the graph is smooth. (This informal way of extending f to the irrational numbers will be suitable for our purposes. A rigorous treatment requires the notion of a limit.) For any positive real number a, we have an exponential function f given by f (x) = ax . Here are the graphs of f for a few choices of a: In general, the graph falls from left to right if a < 1 and it rises from left to right if a > 1. The base e. The most common choice for the base a is a certain number e ≈ 2.71828. The reason for this choice is that it allows simplification of several formulas involving exponential functions. The function f (x) = ex is often called the exponential function. Since e > 1, the graph of this exponential function rises from left to right. It 3 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 3 passes through the y-axis at 1 (as do the graphs of all the exponential functions), and it passes through the point (1, e): 3.3 Logarithm The notation log2 8 can be thought of as posing a question: What power do you raise 2 to in order to get 8? In symbols, log2 8 =? ⇐⇒ 2? = 8. The answer is 3, that is, log2 8 = 3. In general, for any positive real numbers a and y with a 6= 1, the notation loga y stands for the power that a is raised to in order to get y; it is the logarithm with respect to the base a of y (or just “log base a of y”) : loga y = x ⇐⇒ ax = y. (1) From this relationship between the logarithm and the exponential and the laws of exponents, we get the following identities: Laws of logarithms. For any positive real numbers x and y and any real number r, (i) loga (xy) = loga x + loga y, (ii) loga (x/y) = loga x − loga y, (iii) loga xr = r loga x, 3.3.1 Example Solution Express log5 72 in terms of log5 2 and log5 3. We have log5 72 = log5 (23 32 ) = log5 23 + log5 32 = 3 log5 2 + 2 log5 3. 3 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 4 The following identities follow immediately from (1): Inverse properties of logarithm. (i) loga ax = x (x ∈ R), (ii) aloga x = x (x > 0). These identities can be used to solve equations involving exponentials or logarithms as the next example shows. 3.3.2 Example (a) Solve 32x+1 = 4 for x. (b) Solve log5 (1 − x) = 2 for x. Solution (a) Applying log3 to both sides and then using the identity (i) above, we have 32x+1 = 4 =⇒ log3 32x+1 = log3 4 =⇒ 2x + 1 = log3 4 1 x = (log3 4 − 1) 2 =⇒ (b) Raising 5 to both sides and then using the identity (ii), we have log5 (1 − x) = 2 =⇒ =⇒ =⇒ 3.4 5log5 (1−x) = 52 1 − x = 25 x = −24 Logarithmic function The graph of the function g(x) = log2 x can be sketched by plotting a few points and connecting: 3 EXPONENTIAL AND LOGARITHMIC FUNCTIONS x 1/4 1/2 1 2 4 5 g(x) = log2 x −2 −1 0 1 2 We have used for inputs only positive numbers since log2 x is not defined when x is ≤ 0 (there is nothing you can raise 2 to in order to get a number ≤ 0). This graph is the reflection across the 45◦ line y = x of the graph of f (x) = 2x (see 3.2). We know from 2.5 that the graphs of a function and its inverse are always related in this way, so we might suppose that g is the inverse of f . This is in fact the case, since, using (1) of 3.3, we get g(y) = x ⇐⇒ log2 y = x ⇐⇒ 2x = y ⇐⇒ f (x) = y, so g is f −1 (see (1) of 2.5). More generally, for any positive real number a, the logarithmic function g(x) = loga x is the inverse of the exponential function f (x) = ax , so the graphs of these two functions are related as just described. In particular, if a > 1, then the graph of g rises from left to right; if 0 < a < 1, it falls from left to right (cf. 3.2). Natural logarithmic function. Of particular importance for us is the natural logarithmic function g(x) = ln x, where ln means loge . Its graph is the reflection across the 45◦ line y = x of the graph of the exponential function f (x) = ex : 3 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 6 3 – Exercises 3–1 Simplify the following expressions: (a) 16−3/4 (b) 81/6 83/2 (c) log3 1 27 (d) e4 ln 3 3–2 Sketch the graph of the equation y = 2 − ex+1 in steps: (a) Sketch the graph of y = ex+1 . (Hint: The graph is a shifted version of the graph of y = ex shown in 3.2.) (b) Sketch the graph of y = −ex+1 . (Hint: The graph is a reflected version of the graph of y = ex+1 .) (c) Sketch the graph of y = 2 − ex+1 . (Hint: The graph is a shifted version of the graph of y = −ex+1 .) 3–3 2 (a) Solve e1−x = 2 for x. (b) Solve log5 (x − 4) + log5 x = 1 for x. 3–4 Show that any logarithm can be expressed in terms of natural logarithms by establishing the formula ln x loga x = . ln a Hint: Write y = loga x in the corresponding exponential form, apply ln to both sides, and solve for y. 3–5 Suppose that the expression 21/3 has not yet been defined. Give an argument 1/3 similar to those given at the end is forced by √ of Section 3.1 to show that 2 the laws of exponents to equal 3 2.